The expression “linear search in sorted lists” itself sounds strange. Why should we use this algorithm for sorted lists when there are lots of other algorithms that are far more effective? As I mentioned in

my previous post the sequential search is very ineffective in most of the cases and it is primary used for unordered lists. Indeed sometimes it is more useful first to sort the data and then use a faster algorithm like the binary search. On the other hand the analysis shows that for lists with less than ten items the linear search is much faster than the binary search. Although, for instance, binary search is more effective on sorted lists, sequential search can be a better solution in some specific cases with minor changes. The problem is that when developers hear the expression “sorted list” they directly choose an algorithm different from the linear search. Perhaps the problem lays in the way we understand what an ordered list is?

What is a sorted list?

We used to think that this list (1, 1, 2, 3, 5, 8, 13) is sorted. Actually we think so because it is … sorted, but the list (3, 13, 1, 3, 3.14, 1.5, -1) is also sorted, except that we don’t know how. Thus we can think that any array is sorted, although it is not always obvious how. There are basically two cases when sequential search can be very useful. First when the list is very short or when we know in advance that there are some values that are very frequently searched. Let’s say we have a very large list, with hundreds of thousands of items, but actually most of the searches in that list always find the same ten values. This additional information tells us that using a binary search will be quite ineffective in this case. A possible approach, of course, is to place those values at the front of the list and to perform a sequential search.

Self-Organization

Self-organization practically means that every time we search and find the desired value, we somehow change the list so the next search will be far more effective. There are basically two approaches to do that.

1. To move the item one position forward to the front of the list;
2. To move the item directly at the front of the list;

Of course it depends on your case which approach you’ll choose, but it is assumed that the second option, the one that we choose to move the item directly at the front of the list, is better. Indeed if we choose the first option and the list is (…, 24, 31) after constantly searching for those two values the array will be changing from (…, 24, 31) to (…, 31, 24) and once again to (…, 24, 31) and so on and so on. Thus a better solution is to move the desired item directly to the front of the list. Now if we look for the value of “5” in the list

(1, 2, 4, …, 5, …, 398) it will become (5, 1, 2, …, 398) after the value is found. By choosing this approach we can be sure that as the number of searches increases, the most frequently searched values are placed at the front of the list. Now the sequential search is quite a good solution! Here’s an example of sequential search from my previous article. The only change is that after we find the desired value we need to move it to the front of the list.

Application

Using sequential search in sorted lists can be very useful and fast, the only thing is that we need to know in advance that there are some values that are frequently searched. A typical example of this case is the contact list on your phone. Perhaps you have lots of names in there, but most of the times you search in it is to find your best friends’ and family phone numbers. That is why most of the cell phone manufacturers add to their phones the ability to predefine shortcut keys for the most frequently dialed numbers. Here’s another use case. Let’s say that we have the same scenario as in my previous

post, where username/name pairs are stored into a CSV file. We can fetch those values in a PHP array.

Every time a user enters the site we search for his name by his username and a welcome message is displayed. We know that some users enter the site very frequently while others do that once per month so we cannot only perform a sequential search but also we can use self-organization for the array and change the CSV file at the end.

The result is:

Hello, Darth Vader 
Found after 5 iterations!
Hello, Darth Vader
Found after 1 iterations!

Now every time Darth Vader tries to sign in, you won’t bother him to wait a lot for sure. However I bet nobody uses CSV files to store such information, but this is only an example.

This is the easiest to implement and the most frequently used search algorithm in practice. Unfortunately the sequential search is also the most ineffective searching algorithm. However, it is so commonly used that it is appropriate to consider several ways to optimize it. In general the sequential search, also called linear search, is the method of consecutively check every value in a list until we find the desired one.

Basic Implementation

The most natural approach is to loop through the list until we find the desired value. Here’s an implementation on PHP using FOR loop, something that can be easily written into any other computer language.

This is really the most ineffective implementation. There are two big mistakes in this code. First of all we calculate the length of the list on every iteration of the array, and secondly after we find the desired element, we don’t break the loop, but continue to loop through the array.

Yes, if the element is repeated without the “break” we can find its last occurrence, but if not the loop will iterate over the end of the array with no practical value.

Optimization of the forward sequential search

… and javascript:

Even with this little optimization the algorithm remains ineffective. As we can see, on every iteration we have two conditional expressions. First we check whether we’ve reached the end of the list, and then we check whether the current element equals to the searched element. So the question is can we reduce the number of the conditional expressions?

Searching in reverse order

Yes, we can reduce the number of comparison instructions from the forward approach of the linear search algorithm by using reverse order searching. Although it seems to be pretty much the same by reversing the order of the search we can discard one of the conditional expressions.

Note that we need to adjust index because of $index—expression.

Indeed here we have only one conditional expression, but the problem is that this implementation is correct ONLY when the element exists in the list, which is not always true. If the element doesn’t appears into the list, then this code can lead to an infinite loop. OK, but how can we stop the loop even when the list doesn’t contain the desired value? The answer is, by adding the searched value to the list.

Sentinel

The above problem can be solved by inserting the desired item as a sentinel value. Thus we’re sure that the list contains the value, so the loop will stop for sure even if at the beginning the value didn’t appear to be part of the list.

This approach can be used to overcome the problem of the reverse linear search approach from the previous section.

Complexity

As I said at the beginning of this post this is one of the most ineffective searching algorithms. Of course the best case is when the searched value is at the very beginning of the list. Thus on the first comparison we can find it. On the other hand the worst case is when the element is located at the very end of the list. Assuming that we don’t know where the element is and the possibility to be anywhere in the list is absolutely equal, then the complexity of this algorithm is O(n).

Different cases

We must remember, however, that the algorithm’s complexity can vary depending on whether the element occurs once.

Is it so ineffective?

Sequential search can be very slow compared to binary search on an ordered list. But actually this is not quite true. Sequential search can be faster than binary search for small arrays, but it is assumed that for n < 8 the sequential search is faster.

Application

The linear search is really very simple to implement and most web developers go to the forward implementation, which is the most ineffective one. On the other hand this algorithm is quite useful when we search in an unordered list. Yes, searching in an ordered list is something that can dramatically change the search algorithm. Actually searching and sorting algorithms are often used together.

A typical case is pulling something from a database, usually in form of a list and then search for some value in it. Unfortunately in most of the cases the database orders the returned result set and yet most of the developers perform a consecutive search over the list. Yet again when the list is ordered it is better to use binary search instead of sequential search.
Let’s say we have a CSV file containing the usernames and the names of our users.

Username,Name
jamesbond007,James Bond
jsmith,John Smith
...

Now we fetch these values into an array.

// work case
$arr = array(
    array('name' => 'James Bond', 'username' => 'jamesbond007'),
    array('name' => 'John Smith', 'username' => 'jsmith')
);

Now using sequential search …

// using a sentinel
$x = 'jsmith';
$arr[] = array('username' => $x, 'name' => '');
$index = 0;

while ($arr[$index++]['username'] != $x);

if ($index < count($arr)) {
    echo "Hello, {$arr[$index-1]['name']}";
} else {
    echo "Hi, guest!";
}